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Identification and Estimation in Fuzzy Regression Discontinuity Designs with Covariates

Carolina Caetano, Gregorio Caetano, Juan Carlos Escanciano

TL;DR

This paper develops a unified framework for identification and estimation of WLATEs in fuzzy RDDs with covariates, showing that a broad class of point-identified WLATEs can be represented as Wald ratios using covariate-derived instruments. It identifies the Compliance-Weighted LATE (CWLATE) as the WLATE with maximal alignment to the first-stage information, expressed as $\beta_{CW} = \frac{\mathbb{E}[\delta_X(W)\delta_Y(W)]}{\mathbb{E}[\delta_X(W)^2]}$, and provides simple plug-in estimators with robust bias-corrected inference for discrete covariates. The authors develop a local-linear, stacked estimation approach, a delta-method expansion, and RBC methods, along with MSE-optimal bandwidth selectors, enabling practical CWLATE implementation within standard RDD toolkits. Monte Carlo simulations show that CWLATE improves stability and often lowers MSE relative to standard fuzzy RDD estimators when compliance varies by covariates, and an empirical application to Uruguay’s cash-transfer program demonstrates precise, substantively meaningful effects on low birthweight among compliers. Overall, CWLATE offers a credible, more informative estimand when first-stage variation is heterogeneous, enhancing policy-relevant conclusions in fuzzy RDDs with covariates.

Abstract

We study fuzzy regression discontinuity designs with covariates and characterize the weighted averages of conditional local average treatment effects (WLATEs) that are point identified. Any identified WLATE equals a Wald ratio of conditional reduced-form and first-stage discontinuities. We highlight the Compliance-Weighted LATE (CWLATE), which weights cells by squared first-stage discontinuities and maximizes first-stage strength. For discrete covariates, we provide simple estimators and robust bias-corrected inference. In simulations calibrated to common designs, CWLATE improves stability and reduces mean squared error relative to standard fuzzy RDD estimators when compliance varies. An application to Uruguayan cash transfers during pregnancy yields precise RDD-based effects on low birthweight.

Identification and Estimation in Fuzzy Regression Discontinuity Designs with Covariates

TL;DR

This paper develops a unified framework for identification and estimation of WLATEs in fuzzy RDDs with covariates, showing that a broad class of point-identified WLATEs can be represented as Wald ratios using covariate-derived instruments. It identifies the Compliance-Weighted LATE (CWLATE) as the WLATE with maximal alignment to the first-stage information, expressed as , and provides simple plug-in estimators with robust bias-corrected inference for discrete covariates. The authors develop a local-linear, stacked estimation approach, a delta-method expansion, and RBC methods, along with MSE-optimal bandwidth selectors, enabling practical CWLATE implementation within standard RDD toolkits. Monte Carlo simulations show that CWLATE improves stability and often lowers MSE relative to standard fuzzy RDD estimators when compliance varies by covariates, and an empirical application to Uruguay’s cash-transfer program demonstrates precise, substantively meaningful effects on low birthweight among compliers. Overall, CWLATE offers a credible, more informative estimand when first-stage variation is heterogeneous, enhancing policy-relevant conclusions in fuzzy RDDs with covariates.

Abstract

We study fuzzy regression discontinuity designs with covariates and characterize the weighted averages of conditional local average treatment effects (WLATEs) that are point identified. Any identified WLATE equals a Wald ratio of conditional reduced-form and first-stage discontinuities. We highlight the Compliance-Weighted LATE (CWLATE), which weights cells by squared first-stage discontinuities and maximizes first-stage strength. For discrete covariates, we provide simple estimators and robust bias-corrected inference. In simulations calibrated to common designs, CWLATE improves stability and reduces mean squared error relative to standard fuzzy RDD estimators when compliance varies. An application to Uruguayan cash transfers during pregnancy yields precise RDD-based effects on low birthweight.
Paper Structure (27 sections, 30 theorems, 222 equations, 6 figures, 7 tables)

This paper contains 27 sections, 30 theorems, 222 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Identification in RDD with Covariates
  3. Model Setup
  4. Conditional LATEs and WLATEs
  5. Examples
  6. Compliance-Weighted Identification
  7. WLATEs as Expected Policy Effects
  8. Estimation of the CWLATE
  9. Plug-in estimator.
  10. Local linear implementation (stacked form).
  11. Delta-method expansion.
  12. Monte Carlo Simulations
  13. Simulation Setup
  14. Results
  15. Empirical Application
  16. ...and 12 more sections

Key Result

Theorem 2.1

(a) Under Assumption ass:rdd: for any $w\in\mathcal{W}$, $\delta_Y(w)=\beta(w)\,\delta_X(w)$. (b) The conditional LATE $\beta(w)$ is point identified iff (i) $\delta_X(w)\neq0$ and (ii) $w\in\mathcal{W}_{z_0}$, in which case $\beta(w)=\delta_Y(w)/\delta_X(w)$. (c) A WLATE $\beta_\omega=\mathbb{E}[\o The weights, then, can be written as $\omega(W_i)={b(W_i)\,\delta_X(W_i)}/{\mathbb{E}[b(W_i)\,\delt

Figures (6)

  • Figure 1: First-Stage Plots
  • Figure 2: Reduced Form Plots
  • Figure 3: Conditional Plots ($\alpha_{DW}=1,\ \beta_{XW}=2$)
  • Figure 4: Unconditional First-Stage
  • Figure 5: First-Stage by $W_i$
  • ...and 1 more figures

Theorems & Definitions (39)

  • Theorem 2.1: Identification of Conditional LATEs and WLATEs
  • Example 2.1: Unconditional LATE
  • Example 2.2: Average Conditional LATE
  • Example 2.3: Counterfactual WLATE
  • Example 2.4: Maximal Average Social Welfare
  • Theorem 2.2: Compliance-Weighted "Instrument"
  • Theorem 3.1: RBC inference for the CWLATE
  • Theorem A.1
  • Example A.1
  • Example A.2
  • ...and 29 more